Gauge fixing

Remarks

A gauge condition is complete if the gauge transformation that achieves it is unique. Most usual conditions leave a residual freedom, for example by functions $\chi$ that solve a homogeneous equation. In covariant notation the transformation reads $A^\mu \mapsto A^\mu - \partial^\mu \chi$ for the four-potential.

Example: the Coulomb gauge is $\nabla \cdot A = 0$. Given any $(\phi, A)$ on $\R^3$ with $\nabla \cdot A$ smooth and decaying, solve the Poisson equation $\Delta \chi = -\nabla \cdot A$; then $\nabla \cdot (A + \nabla \chi) = 0$.

Other common conditions are the Lorenz gauge $\nabla \cdot A + \frac{1}{c^2} \partial_t \phi = 0$, the temporal gauge $\phi = 0$ and the axial gauge $A_3 = 0$. For the temporal gauge choose $\chi(x, t) = \int_0^t \phi(x, s) \, ds$, which gives $\phi - \partial_t \chi = 0$.

Non-example: the condition $A = 0$ is not a gauge condition, since it forces $B = 0$ and so cannot be reached from potentials of a nonzero magnetic field.